EXPANDING AND CONTRACTING UNIVERSES IN THIRD QUANTIZED STRING COSMOLOGY

CERN-TH/96-322 IFUP-TH/96-65

hep-th/9701146

X. Xxxxxxxx(a)(b), X. Xxxxxxxxx(c)(d), X. Xxxxxxxx(a)(b) and X. Xxxxxxxxx(a)(b)

(a) Dipartimento di Fisica, Universita` di Pisa,

Xxxxxx Xxxxxxxxxx 0, X-00000 Xxxx, Xxxxx

(b) Istituto Nazionale di Fisica Nucleare, Sezione di Pisa, Pisa, Italy

(d) Dipartimento di Fisica Teorica, Universit`a di Torino, Xxx X. Xxxxxx 0, 00000 Xxxxx, Xxxxx

Abstract

We discuss the possibility of quantum transitions from the string perturbative vacuum to cosmological configurations characterized by isotropic contraction and decreasing dilaton. When the dilaton potential preserves the sign of the Hubble factor throughout the evolution, such transitions can be represented as an anti-tunnelling of the Xxxxxxx–De Xxxx wave function in minisuperspace or, in a third-quantization language, as the production of pairs of universes out of the vacuum.

To appear in Class. Quantum Grav.

CERN-TH/96-322

November 1996

At very early times, according to the standard cosmological scenario, the Universe is ex- pected to approach a Planckian, quantum gravity regime where a classical description of the spacetime manifold is no longer appropriate. A possible quantum description of the Universe, in that regime, is based on the Xxxxxxx–Xx Xxxx (WDW) wave function [1, 2], generally de- fined on the superspace spanned by all three-dimensional geometric configurations. In that context it becomes possible to compute, with an appropriate model of (mini)superspace, the probability distribution of a given cosmological configuration versus an appropriate “state” parameter (for instance the cosmological constant Λ). The results, however, are in general affected by operator-ordering ambiguities, and are also strongly dependent on the boundary conditions [3]–[5] imposed on the solutions of the WDW equation.

String theory has recently motivated the study of a cosmological scenario in which the Universe starts from the string perturbative vacuum and evolves through an initial, “pre-big bang” phase [6], characterized by an accelerated growth of the curvature and of the gauge coupling g = eφ/2 (φ is the dilaton field). In such a context, the WDW equation is obtained from the low-energy string effective action [7]–[9], and has no operator ordering ambiguities

[7] since the ordering is uniquely fixed by the duality symmetries of the action. Also, the boundary conditions are determined by the choice of the perturbative vacuum as the initial state for the cosmological evolution.

According to the lowest-order effective action, the classical evolution from the pertur- bative vacuum necessarily leads the background to a singularity, and the transition to the present decelerated “post-big bang” configuration is impossible, for any realistic type of (local) dilaton potential [10]. With an appropriate potential, however, the transition may become allowed at the quantum level even if, for the same potential, it remains classically forbidden. This effect was discussed in previous papers [7], in which the WDW equation was applied to compute the transition probability between two duality-related pre- and post-big bang cosmological phases.

The string perturbative vacuum is, in general, a higher-dimensional state, and the initial growth of the dilatonic coupling g requires, according to the lowest-order action, a large enough number of expanding dimensions. For instance, in a Bianchi-type I background with

d expanding and n contracting isotropic spatial dimensions, the growth of g requires [6]

√

d + d + n > n, which cannot be satisfied by d = 3, in particular, in the ten-dimensional

superstring vacuum. With a monotonic evolution of the scale factor, this represents another obstruction to a smooth transition to our present, dimensionally reduced Universe.

The aim of this paper is to show that the initial perturbative vacuum is not inconsistent, at the quantum level, with a final contracting cosmological configuration, when we add to the lowest-order action an appropriate dilaton potential (such as the simple one induced by an effective cosmological constant). In particular, for a WDW potential which is translationally invariant in minisuperspace, along the direction parametrized by the scale factor, and for which the sign of the Hubble factor is classically conserved during the whole evolution, the cosmological contraction corresponds to a pure quantum effect. It can be described as an “anti-tunnelling” of the WDW wave function from the string perturbative vacuum or, in a third quantization [11] language, as a production of “pairs of universes” (one expanding, the other contracting) out of the third quantized vacuum. Such a process requires the identification of the time-like coordinate in minisuperspace with the direction parametrized by the shifted dilaton φ (see below), and is complementary to the process of spatial reflection of the wave function, which describes transitions from pre- to post-big bang configurations [7].

We shall adopt, in this paper, the minisuperspace model already discussed in [7], based on the tree-level, lowest-order in αj, string effective action [12]. Working in the simplifying assumption that only the metric and the dilaton contribute non-trivially to the background,

in d isotropic spatial dimensions, the corresponding action can be written as

S = − 2 λd—1

∫ dd+1x √|g| e—φ (R + ∂µ

φ∂µφ + V ) . (1)

Here λs = (αj)1/2 is the fundamental string length parameter governing the higher-derivative expansion of the action, and V is a (possibly non-perturbative) dilaton potential. By using the parametrization appropriate to an isotropic, spatially flat cosmological background:

√

gµν = diag N2(t), −a2(t)δij , a = exp hβ(t)/ di , φ = φ(t), (2)

and assuming spatial sections of finite volume, the action can be expressed in the convenient

∫

form

λs

S =

where φ is the shifted dilaton:

e—φ

β˙2

− ˙2

− N V

, (3)

dβ .

φ = φ − log ∫ ddx/λd − √

(4)

The variation with respect to N then leads to the Hamiltonian constraint

Π2 − Π2 + λ2 V (β, φ) e—2 φ = 0 , (5)

β φ s

where Πβ, Πφ are the (dimensionless) canonical momenta (in the gauge N = 1):

δS

Πβ = δβ˙

= λs

β˙ e—φ, Π

δS

= δφ˙

= −λs

φ˙ e—φ. (6)

When V = 0, the classical solutions of the action (3) describing the phase of accelerated pre-big bang evolution are characterized by two duality-related branches [6], defined in the negative time range:

√

t < 0, a = a0(−t)∓1/

d, φ − φ0 = − ln(−t) = ±β, Πβ = ±k = const, Π

= ∓Πβ < 0

(7)

(k, a0 and φ0 are integration constants). For the upper-sign branch the metric is expanding

(Πβ > 0), and the curvature scale

β˙2 and the string coupling g(t) are growing, starting

asymptotically from the perturbative vacuum, the state with flat metric (β˙

= 0 = φ˙) and

vanishing coupling constant (φ = −∞, g = 0). The lower-sign branch corresponds instead to a contracting configuration (Πβ < 0), in which the coupling g(t) is decreasing. In the presence of a constant dilaton potential, V = Λ = const, the accelerated pre-big bang

solutions are again characterized by two branches [13]:

√ √ √

t < 0, a = a0

htanh(−t Λ/2)i∓1/ d , φ − φ = − ln sinh −t Λ , Π

0 φ

< 0, (8)

which are respectively expanding with growing dilaton (upper sign, Πβ > 0) and contracting with decreasing dilaton (lower sign, Πβ < 0). In this case both branches are dominated, in the low-curvature regime, by the contribution of a positive cosmological constant Λ. The initial perturbative vacuum is replaced by a configuration with flat metric and linearly

evolving dilaton (β˙

= 0, φ˙

= const), another well-known string theory background [14]

(exact solution to all orders in the αj expansion). Near the singularity (t → 0—), however, the contribution of Λ becomes negligible, and the solution (8) asymptotically approaches that of eq. (7).

In this paper we shall assume that an effective cosmological constant Λ is generated non- perturbatively in the strong coupling, Planckian regime, and we shall use the WDW equation to discuss the possibility of transitions, induced by Λ, from the perturbative vacuum to a

final configuration with contracting metric and decreasing dilaton. We shall consider, in particular, the case in which the effective dilaton potential can be approximated by the Heaviside step function θ as V (β, φ) = Λ θ(φ). The corresponding WDW equation, in the minisuperspace spanned by β and φ, is obtained from the Hamiltonian constraint (5) through the differential representation Π = −i∇:

h∂2 − ∂2 + λ2 Λ θ(φ) e—2φi Ψ = 0 . (9) The momentum along the β axis is conserved,

[Πβ, H] = 0, Πβ = λsβ˙e—φ = k = const, (10)

and the general solution of the WDW equation can be factorized as Ψk(φ, β) = fik(φ)eikβ.

Note that we have assumed a potential V depending explicitly only on φ because the classical evolution of the scale factor, in that case, is monotonic, and no contracting config- uration can be eventually obtained, classically, if we start from the isotropic perturbative vacuum. From a quantum-mechanic point of view, however, the situation is different. In- deed, if we assign to φ the role of time-like coordinate, eq. (9) is formally equivalent to a Xxxxx–Xxxxxx equation with time-dependent mass term. The solution fik is a linear com-

bination of plane waves for φ < 0, and of Bessel functions [15] J±ν(z), of imaginary index

√

ν = ik and argument z = λs Λe—φ, for φ > 0. In particular, the functions

(±)

eikβ

∓ikφ

Ψk = √ e

Ψk = √

4πk

Γ(1 ± ν) J±ν(z) , φ > 0, (12)

4πk

, φ < 0, (11)

(±)

eikβ

z0 ∓ν

√

where z0 = λs Λ and Γ is the Euler function, provide orthonormal sets of solutions with

respect to the Xxxxx-Xxxxxx scalar product

∂φ

(Ψ1, Ψ2) = −i ∫ dβ Ψ1(β, φ) ↔

Ψ2∗(β, φ) . (13)

We shall fix the boundary conditions by imposing that, for φ < 0, the Universe is

represented by the wave function

ik(β—φ)

Ψ (β, φ < 0) = √ e , (14)

4πk

corresponding to a state of growing dilaton and accelerated pre-big bang expansion from the perturbative vacuum, with Πβ = −Πφ = k > 0 according to eq. (7). The eigenvalue k of Πβ parametrizes the initial state in the space of all classical configurations (7). For φ > 0 the

wave function is uniquely determined by the matching conditions for Ψ and ∂φΨ at φ = 0, in terms of the functions (12), as

ΨIIk(β, φ > 0) = A+ Ψ(+) + A— Ψ(—), (15)

k k k k

Γ(1 ∓ ik)

±J∓ik(z0) ∓

J∓ik(z0)

(16)

where

Ak =

± iz0

2 k

z0 ±ik

" j ik #

(a prime denotes differentiation of the Bessel functions with respect to their argument).

Given a pure initial state Ψ(+)

of “positive frequency” k, the final state is thus a mixture

of “positive” and “negative” frequency modes, Ψ(+) and Ψ(—), satisfying asymptotically the

conditions

II II

lim Ψ(±)(β, φ) = Ψ(±)(β, φ) ∼ eik(β∓φ),

II ∞

φ→∞

ΠβΨ(±) = −i∂βΨ(±) = kΨ(±), Π Ψ(±) = −i∂ Ψ(±) = ∓ΠβΨ(±). (17)

∞ ∞ ∞ φ ∞ φ ∞ ∞

The mixing is determined by the coefficients A±, satisfying the standard Bogoliubov normalization condition |A+|2 −|A—|2 = 1. In a second quantization context, it is well known

k k

that such a mixing describes a process of pair production [16], the negative energy mode being associated to an antiparticle state of positive energy and opposite spatial momentum. It thus seems correct to interpret the above splitting of the WDW wave function, in a third quantization context [11], as the production of a pair of universes, with quantum numbers

{Πβ, Πφ}, corresponding to positive energy (Πφ < 0) and opposite momentum along the

spacelike direction β. One of the two universes is isotropically expanding (Πβ > 0), with growing dilaton; the “anti-universe” is isotropically contracting (Πβ < 0), with decreasing dilaton. Both configurations evolve towards the curvature singularity of the classical pre-big bang solution (8). However, while the growing dilaton state corresponds to a continuous classical evolution from the perturbative vacuum, no smooth connection to such vacuum is possible, classically, for the state with decreasing dilaton.

It is important to stress that, as long as V = V (φ) and, consequently, Πβ is conserved, a third-quantized production of universes is only possible provided we assign the role of

time-like coordinate to φ, and the potential satisfies V (φ)e—2φ → 0 for φ → +∞ (in order to identify, asymptotically, positive and negative frequency modes). The pairs of universes are produced in the limit of large positive φ, so that we cannot describe in this context

a transition to post-big bang cosmological configurations, which are instead characterized by φ < 0. A quantum description of the transition from pre- to post-big bang requires in fact the interpretation of β as the time-like axis, as discussed in [7]. In that case, a third quantized production of pairs becomes possible only if Πβ is not conserved, namely if V depends also on β.

For the process considered in this paper, the probability is controlled by |A—|2, which

determines the expectation number pairs of universes produced in the final state. The

production probability is negligible when |A—| 1; it has the typical probability of a vacuum fluctuation effect when |A—|∼ |A+|∼ 1; finally, when |A—|∼ |A+| 1, the initial

k k k k

wave function is parametrically amplified [17] and the probability is large. In our case, the interesting parameter characterizing the process, besides Λ, is the portion of proper spatial volume Ω = ad ddx undergoing the transition. Considering, in particular, d = 3 spatial

dimensions, and using the definitions of k and φ, the initial momentum k can be conveniently

√

expressed as k = 3Ωsg—2λ—3, where gs = exp(φs/2) and Ωs are, respectively, the value of

s s

the coupling and of the proper spatial volume evaluated at the string scale t = ts, when

√

H ≡ β˙/ 3 = λ—1. By exploiting the properties of the Bessel functions, we can then express

the asymptotic limits of the Bogoliubov coefficients (16) in terms of the physical parameters

Ωs and Λ. We obtain, at fixed Ωs/(g2λ3) = 1,

s s

|A+|2 − 1 ' |A—|2 ' 1 Λ2 λ4 , Λ λ—2, (18)

48 s s

cosh(

3 π) − sin(2

Λ λs)

' |A

Λ λs

√

√ √ 2

+ 2

— 2

—2

√ √ , Λ λs , (19)

and, at fixed Λλ2 = 1,

g4 λ6

4 3 sinh(

3 π)

|A+|2 ' |A—|2 ' s s |Jj (1)|2 , |Jj (1)|' 0.44, Ωs g2λ3

(20)

12 Ω2 0 0 s s

(the limit Ωs g2λ3 cannot be performed because the quantum process is confined to the

s s

region of large φ).

The quantum production of universes in a state with non-vanishing cosmological constant Λ is thus strongly suppressed for small values of Λ, while it is favoured in the opposite limit

of large Λ and of proper volumes that are small in string units, in qualitative agreement with previous results [7], and also with the general approach to quantum cosmology based on tunnelling boundary conditions [4, 5]. Instead of a “tunnelling from nothing”, however, this quantum production of expanding and contracting universes can be seen as an “anti- tunnelling from the string perturbative vacuum” of the WDW wave function. Indeed, the asymptotic expansion of the solution (14), (15),

φ → +∞ , fi ∼ Aine—ikφ + Xxxx xxxx,

φ → −∞ , fi ∼ Atre—ikφ, (21)

describes formally a scattering process along φ, in which the expanding universe corresponds to the incident part of the wave function, the contracting anti-universe to the reflected part, and the initial vacuum to the transmitted part. In the parametric amplification regime of eqs. (19) and (20), where |A+| ∼ |A—| 1, the reflection coefficient R = |Aref |2/|Ain|2 is

approximately 1, and the Bogoliubov coefficient |A—|, which controls the probability of pair

production, becomes the inverse of the tunnelling coefficient T = |Atr|2/|Ain|2:

— 2 |Aref |2 R 1

|A |

|Atr

|2 = T ' T . (22)

In view of future applications, we have also computed numerically the Bogoliubov coef- ficients A± by discretizing the WDW equation with the explicit method [18], and using the routine Fast Fourier Transform [19]. A computer simulation, in which the pair production process is graphically represented by the scattering and reflection of an initial wave packet, has given results in complete agreement with the analytic computation (16).

In conclusion, we have shown in this paper that it is not impossible, in a quantum cosmology context, to nucleate universes in a state characterized by isotropic contraction and decreasing dilaton. The process can be described as the production from the vacuum of universe–anti-universe pairs in the strong coupling regime, triggered by the presence of an effective cosmological constant. When V = V (φ) and Πβ is conserved, the pair-production process requires the identification of φ as time-like coordinate in minisuperspace, while the transition from pre- to post-big bang configurations requires the complementary choice of β as the time-like axis.

The validity of our analysis is limited by the very crude approximation (the step potential) adopted to modellize the time-evolution of the non-perturbative dilaton potential. Also, an

appropriate potential should depend on φ (not on φ as assumed in this paper); in that case, however, the transition from expansion to contraction may be allowed also classically (in an appropriate limit), and is represented in minisuperspace as a reflection [20] (instead of an anti-tunnelling) of the wave function. In spite of these limitations, the analysis of this paper confirms that the WDW approach provides an adequate framework for a consistent formulation of quantum string cosmology, with the boundary conditions uniquely prescribed by the choice of the initial perturbative vacuum.

Acknowledgements

We are grateful to Xxxxxxxx xx Xxxxxx and Xxxxxxx Xxxxx for discussions and clarifying com- ments. Special thanks are due to Xxxxxxxx Xxxxxxxxx for a careful reading of the manuscript and for helpful suggestions.

References

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[18] Xxxxxxxx A R and Xxxxxxxxx X X 1980 The finite difference method in partial differential equations (Xxxx Xxxxx and Sons Ltd., Chichester).

[19] Press W H et al 1992 Numerical recipes in Fortran: the art of scientific computing

(Cambridge University Press, Cambridge).

[20] Xxxxx R et al 1996 in preparation.

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